How to make operator data persistent?
There are a few options open to you:


Yes you can keep the operator instance by keeping the operator running, in this case you would need to start a modal operator, then use some action to trigger it (think the knife tool - it stays running and but act's when the mouse is dragged).Note: This is the most direct answer to your question - however its only really useful in specific cases.
Store the settings externally.
Typically when you store settings externally, the operators invoke function will read from these settings when initializing.


In a Python dictionary outside the operator instance (settings will be lost on restarting Blender unless you implement callbacks to write and restore them)
In custom properties (custom Scene properties is often used for this)
In the addons user preferences (not so good for per-blend-file data)
In a text data block (normally not optimal but including for completeness)